CS 326 A: Motion Planning robotics.stanford.edu/~latombe/cs326/2003/index.htm Configuration Space – Basic Path-Planning Methods

What is a Path?

Tool: Configuration Space (C-Space C)

q=(q 1,…,q n ) Configuration Space q1q1q1q1 q2q2q2q2 q3q3q3q3 qnqnqnqn

Definition A robot configuration is a specification of the positions of all robot points relative to a fixed coordinate system Usually a configuration is expressed as a “vector” of position/orientation parameters

reference point Rigid Robot Example 3-parameter representation: q = (x,y,  ) In a 3-D workspace q would be of the form (x,y,z,  ) x y  robot reference direction workspace

Articulated Robot Example q1q1q1q1 q2q2q2q2 q = (q 1,q 2,…,q 10 )

Configuration Space of a Robot Space of all its possible configurations But the topology of this space is usually not that of a Cartesian space C = S 1 x S 1

Configuration Space of a Robot Space of all its possible configurations But the topology of this space is usually not that of a Cartesian space C = S 1 x S 1

Configuration Space of a Robot Space of all its possible configurations But the topology of this space is usually not that of a Cartesian space C = S 1 x S 1

Structure of Configuration Space It is a manifold For each point q, there is a 1-to-1 map between a neighborhood of q and a Cartesian space R n, where n is the dimension of C This map is a local coordinate system called a chart. C can always be covered by a finite number of charts. Such a set is called an atlas

Example

reference point Case of a Planar Rigid Robot 3-parameter representation: q = (x,y,  ) with   [0,2  ). Two charts are needed Other representation: q = (x,y,cos ,sin  )  c-space is a 3-D cylinder R 2 x S 1 embedded in a 4-D space x y  robot reference direction workspace

Rigid Robot in 3-D Workspace q = (x,y,z,  ) Other representation: q = (x,y,z,r 11,r 12,…,r 33 ) where r 11, r 12, …, r 33 are the elements of rotation matrix R: r 11 r 12 r 13 r 21 r 22 r 23 r 31 r 32 r 33 with: –r i1 2 +r i2 2 +r i3 2 = 1 –r i1 r j1 + r i2 r 2j + r i3 r j3 = 0 –det(R) = +1 The c-space is a 6-D space (manifold) embedded in a 12-D Cartesian space. It is denoted by R 3 xSO(3)

Parameterization of SO(3) Euler angles: (  Unit quaternion: (cos  /2, n 1 sin  /2, n 2 sin  /2, n 3 sin  /2) x yz x yz x y z  x yz 1  2  3  4

Metric in Configuration Space A metric or distance function d in C is a map d: (q 1,q 2 )  C 2  d(q 1,q 2 ) > 0 such that: –d(q 1,q 2 ) = 0 if and only if q 1 = q 2 –d(q 1,q 2 ) = d (q 2,q 1 ) –d(q 1,q 2 ) < d(q 1,q 3 ) + d(q 3,q 2 )

Metric in Configuration Space Example: Robot A and point x of A x(q): location of x in the workspace when A is at configuration q A distance d in C is defined by: d(q,q’) = max x  A ||x(q)-x(q’)|| where ||a - b|| denotes the Euclidean distance between points a and b in the workspace

Specific Examples in R 2 x S 1 q = (x,y,  ), q ’ = (x ’,y ’,  ’ ) with   ’  [0,2  )  = min{ |  ’ |, 2  |  ’ | } d(q,q ’ ) = sqrt[(x-x ’ ) 2 + (y-y ’ ) 2 +  2 ] d(q,q ’ ) = sqrt[(x-x ’ ) 2 + (y-y ’ ) 2 + (  ) 2 ] where  is the maximal distance between the reference point and a robot point ’’’’  

Notion of a Path  A path in C is a piece of continuous curve connecting two configurations q and q’:  : s  [0,1]   (s)  C  s’  s  d(  (s),  (s’))  0 q 1 q 3 q 0 q n q 4 q 2  (s)

Other Possible Constraints on Path  Finite length, smoothness, curvature, etc…  A trajectory is a path parameterized by time:  : t  [0,T]   (t)  C q 1 q 3 q 0 q n q 4 q 2  (s)

Obstacles in C-Space A configuration q is collision-free, or free, if the robot placed at q has null intersection with the obstacles in the workspace The free space F is the set of free configurations A C-obstacle is the set of configurations where the robot collides with a given workspace obstacle A configuration is semi-free if the robot at this configuration touches obstacles without overlap

Disc Robot in 2-D Workspace

Rigid Robot Translating in 2-D CB = B A = {b-a | a  A, b  B} a1 b1 b1-a1

Linear-Time Computation of C-Obstacle in 2-D (convex polygons)

Rigid Robot Translating and Rotating in 2-D

C-Obstacle for Articulated Robot

Free and Semi-Free Paths  A free path lies entirely in the free space F  A semi-free path lies entirely in the semi-free space

Remark on Free-Space Topology The robot and the obstacles are modeled as closed subsets, meaning that they contain their boundaries One can show that the C-obstacles are closed subsets of the configuration space C as well Consequently, the free space F is an open subset of C. Hence, each free configuration is the center of a ball of non-zero radius entirely contained in F The semi-free space is a closed subset of C. Its boundary is a superset of the boundary of F

Notion of Homotopic Paths Two paths with the same endpoints are homotopic if one can be continuously deformed into the other R x S 1 example:  1 and  2 are homotopic  1 and  3 are not homotopic In this example, infinity of homotopy classes q q’   

Connectedness of C-Space C is connected if every two configurations can be connected by a path C is simply-connected if any two paths connecting the same endpoints are homotopic Examples: R 2 or R 3 Otherwise C is multiply-connected Examples: S 1 and SO(3) are multiply- connected: - In S 1, infinity of homotopy classes - In SO(3), only two homotopy classes

Classes of Homotopic Free Paths

Example for Articulated Robot

Motion-Planning Framework Continuous representation (configuration space formulation) Discretization Graph searching (blind, best-first, A*)

Path-Planning Approaches 1.Roadmap Represent the connectivity of the free space by a network of 1-D curves 2.Cell decomposition Decompose the free space into simple cells and represent the connectivity of the free space by the adjacency graph of these cells 3.Potential field Define a function over the free space that has a global minimum at the goal configuration and follow its steepest descent

Roadmap Methods  Visibility graph Introduced in the Shakey project at SRI in the late 60s. Can produce shortest paths in 2- D configuration spaces

Roadmap Methods  Visibility graph  Voronoi diagram Introduced by Computational Geometry researchers. Generate paths that maximizes clearance. Applicable mostly to 2-D configuration spaces

Roadmap Methods  Visibility graph  Voronoi diagram  Silhouette First complete general method that applies to spaces of any dimension and is singly exponential in # of dimensions [Canny, 87]  Probabilistic roadmaps

Path-Planning Approaches 1.Roadmap Represent the connectivity of the free space by a network of 1-D curves 2.Cell decomposition Decompose the free space into simple cells and represent the connectivity of the free space by the adjacency graph of these cells 3.Potential field Define a function over the free space that has a global minimum at the goal configuration and follow its steepest descent

Cell-Decomposition Methods Two families of methods:  Exact cell decomposition The free space F is represented by a collection of non-overlapping cells whose union is exactly F Examples: trapezoidal and cylindrical decompositions

Trapezoidal decomposition

Cell-Decomposition Methods Two families of methods:  Exact cell decomposition  Approximate cell decomposition F is represented by a collection of non- overlapping cells whose union is contained in F Examples: quadtree, octree, 2 n -tree

Octree Decomposition

Path-Planning Approaches 1.Roadmap Represent the connectivity of the free space by a network of 1-D curves 2.Cell decomposition Decompose the free space into simple cells and represent the connectivity of the free space by the adjacency graph of these cells 3.Potential field Define a function over the free space that has a global minimum at the goal configuration and follow its steepest descent

Potential Field Methods Goal Robot  Approach initially proposed for real-time collision avoidance [Khatib, 86]. Hundreds of papers published on it. Path planning: - Regular grid G is placed over C-space - G is searched using a best-first algorithm with potential field as the heuristic function

Potential Field Methods  Approach initially proposed for real-time collision avoidance [Khatib, 86]. Hundreds of papers published on this topic.  Potential field: Scalar function over the free space  Ideal field (navigation function): Smooth, global minimum at the goal, no local minima, grows to infinity near obstacles  Force applied to robot: Negated gradient of the potential field. Always move along that force